LAGUERRE-FREUD EQUATIONS FOR THE RECURRENCE COEFFICIENTS OF SEMICLASSICAL ORTHOGONAL POLYNOMIALS

In this paper we give a new scheme for deriving a non-linear system, s atisfied by the three-term recurrence coefficients of semi-classical o rthogonal Polynomials, this non-linear system is labelled ''Laguerre-F reud's equations.'' Here we do not deal with the numerical aspect of t he question (stability, asymptotic, ...). Our purpose is to take due a dvantage of linear functionals formalism and to show that given a semi -classical linear functional, i.e., given two polynomials, we are able to provide the Laguerre-Freud's equations. The way of obtaining these equations is put in a recursive form appropriate for computer algebra calculation, especially when the degrees of the two given polynomials are large. We illustrate our process by several examples and we point out two cases where the solutions to Laguerre-Freud's equations are n ot unique. (C) 1994 Academic Press, Inc.